+0  
 
0
235
3
avatar

Please provide indepth explination thanks!

Let $\triangle ABC$ be a right triangle, with the point $H$ the foot of the altitude from $C$ to side $\overline{AB}$.

Prove that 

 Oct 29, 2018
 #1
avatar+3994 
+1

Try to expand the terms! 

 Oct 30, 2018
 #2
avatar+972 
+3

Expanding the terms, 

 

\((x^2+2xh+h^2)+(y^2+2yh+y^2)=(a^2+2ab+b^2) \)

 

Using Pythagorean Theorem,

 

\(x^2+h^2=a^2\\ y^2+h^2=b^2\)

 

We could subsitute the values in, and rewrite the equation

 

\(a^2+2xh+b^2+2yh=a^2+2ab+b^2\\ 2xh+2yh=2ab\\ h(x+y)=ab \)

 

\([ABC]=\frac12 AB\cdot CH = \frac12 BC \cdot AC\\ \frac12 (x+y)h=\frac12 ab\\ h(x+y)=ab\)

 

I hope this helped,

 

Gavin

 Oct 30, 2018
edited by GYanggg  Oct 30, 2018
 #3
avatar+98044 
+2

Note that triangle  BCA  is similar to triangle  BHC

 

Which implies that

 

HC / BC  =  CA / BA..... so...

 

h / a  =  b / ( x + y)    (1)

 

Now expand   ( x + h)^2 + ( y + h)^2  =  ( a + b)^2    (2)

 

x^2 + 2xh + h^2  + y^2 + 2yh + h^2  =  a^2 + 2ab + b^2     (3) 

 

And since  x^2 + h^2  =  a^2    and   y^2 + h^2  = b^2

 

We can subtract these equal parts from  (3)  and we are left with

 

2xh + 2yh  =  2ab        divide through by 2

 

xh + yh  =  ab      factor out h on the left

 

h ( x + y)  =  ab       rearrange as

 

h / a  =  b / ( x + y)     but, by (1)....this is true

 

So (2)  must be true, as well

 

cool cool cool

 Oct 31, 2018

11 Online Users

avatar
avatar